Eilenberg–Ganea theorem

In mathematics, particularly in homological algebra and algebraic topology, the Eilenberg–Ganea theorem states for every finitely generated group G with certain conditions on its cohomological dimension (namely \( {\displaystyle 3\leq \operatorname {cd} (G)\leq n}) \), one can construct an aspherical CW complex X of dimension n whose fundamental group is G. The theorem is named after Polish mathematician Samuel Eilenberg and Romanian mathematician Tudor Ganea. The theorem was first published in a short paper in 1957 in the Annals of Mathematics.[1]

Definitions

Group cohomology: Let G be a group and let \( {\displaystyle X=K(G,1)} \) be the corresponding Eilenberg−MacLane space. Then we have the following singular chain complex which is a free resolution of \( \mathbb {Z} \) over the group ring \( {\displaystyle \mathbb {Z} [G]} \) (where \( \mathbb {Z} \) is a trivial \( {\displaystyle \mathbb {Z} [G]} \) -module):

\( {\displaystyle \cdots {\xrightarrow {\delta _{n}+1}}C_{n}(E){\xrightarrow {\delta _{n}}}C_{n-1}(E)\rightarrow \cdots \rightarrow C_{1}(E){\xrightarrow {\delta _{1}}}C_{0}(E){\xrightarrow {\varepsilon }}Z\rightarrow 0,} \)

where E is the universal cover of X and \({\displaystyle C_{k}(E)} \) is the free abelian group generated by the singular k-chains on E. The group cohomology of the group G with coefficient in a \( {\displaystyle \mathbb {Z} [G]} \)-module M is the cohomology of this chain complex with coefficients in M, and is denoted by \( {\displaystyle H^{*}(G,M)}. \)

Cohomological dimension: A group G has cohomological dimension n {\displaystyle n} n with coefficients in Z (denoted by cd \( {\displaystyle \operatorname {cd} _{Z}(G)} \) ) if

\( {\displaystyle n=\sup\{k:{\text{There exists a }}Z[G]{\text{ module }}M{\text{ with }}H^{k}(G,M)\neq 0\}.} \)

Fact: If G has a projective resolution of length at most n, i.e., Z as trivial \( {\displaystyle Z[G]} \) module has a projective resolution of length at most n if and only if \( {\displaystyle H_{Z}^{i}(G,M)=0}\) for all Z-modules M and for all i>n.

Therefore we have an alternative definition of cohomological dimension as follows,

Cohomological dimension of G with coefficient in Z is the smallest n (possibly infinity) such that G has a projective resolution of length n, i.e., Z has a projective resolution of length n as a trivial Z[G] module.

Eilenberg−Ganea theorem

Let G be a finitely presented group and \( n\geq 3 \)be an integer. Suppose the cohomological dimension of G with coefficients in Z is at most n, i.e., cd \( {\displaystyle \operatorname {cd} _{Z}(G)\leq n} \). Then there exists an n-dimensional aspherical CW complex X such that the fundamental group of X is G, i.e., \( {\displaystyle \pi _{1}(X)=G}. \)

Converse

Converse of this theorem is an consequence of cellular homology, and the fact that every free module is projective.

Theorem: Let X be an aspherical n-dimensional CW complex with π1(X) = G, then cdZ(G) ≤ n.